\(\int { sin } \)e^x . d (e^x) = ________+c.

(a)

cos (e^x)

(b)

sin(e^x)

(c)

- cos (e^x)

(d)

-sin(e^x)

\(\int { \frac { { e }^{ x } }{ \left( 1+{ e }^{ x } \right) ^{ 2 } } } \) dx =_______+c

(a)

\(\frac { 1 }{ 1+{ e }^{ x } } \)

(b)

-\(\frac { 1 }{ 1+{ e }^{ x } } \)

(c)

1 + e^x

(d)

\(\frac { \left( 1+{ e }^{ x } \right) ^{ 3 } }{ 3 } \)

\(\int { { tan }^{ 3 } } 2sec2x\) dx = ___________+c.

(a)

\(\frac { 1 }{ 6 } \) sec³ 2x

(b)

\(\frac { 1 }{ 6 } \) sec³2x - \(\frac { 1 }{ 2 } \) sec 2x

(c)

\(\frac { 1 }{ 2 } \) sec 2x

(d)

\(\frac { 1 }{ 6 } \) sec³ 2x + \(\frac { 1 }{ 2 } \) sec 2x

\(\int { \frac { { 4x }^{ 3 }+1 }{ { x }^{ 4 }+x } } \) dx = _______ + c.

(a)

log (4x³  +1)

(b)

log (x⁴ + x)

(c)

log (4x³)

(d)

\(\frac { 1 }{ log({ x }^{ 4 }) } \)

\(\int { \left| x \right| ^{ 3 } } \) dx is equal to ________+c.

(a)

\(\frac { -{ x }^{ 4 } }{ 4 } +c\)

(b)

\(\frac { \left| x \right| ^{ 4 } }{ 4 } \)

(c)

\(\frac { { x }^{ 4 } }{ 4 } \)

(d)

none of these